Jan 10 Dozenal

I've been keeping this article in my back pocket for a while. Dozenal is one of the common names for the base 12 number system. You are currently using base 10 which we call decimal, where you have 10 different digits available to write your numbers with: 0, 1, 2, … 8, 9. Dozenal adds two further digits 𝛘 and Ɛ pronounced Dec (like decade meaning 10 and written as Chi, but matching 𝛘 meaning 10 in Roman Numerals) and El (a shortening of Eleven).

As we count we can get all the way from 1 to Ɛ while keeping in single digits, then just like with decimal the next number in the sequence is written by clocking the units place back around to zero while adding 1 to the next place along, thus the number afterwards Ɛ is 10. Notice that the number 10 represents a 1 in the twelve’s places and is the equivalent of 12 in decimal.

Here's a grid of the first 145 dozenal numbers starting at 0:

So if we take some examples, 47 on the grid is pronounced four do (like a female deer) seven and it represents 4 lots of 12 plus another 7 from the unit’s place = 55 in decimal. 7𝛘 is seven do dec and is equivalent to 7x12+10=94 in decimal.

Once both the unit’s place and the twelve’s place fill up then we have to clock over to the 144’s place (12 squared, just like in decimal we clock over to the 100’s place which is 10 squared), so 243 is 2x144+4×12+3=339 in decimal.

So why would we want to complicate everything like this? Well the choice to use base 10 was just because we had a convenient 10 fingers that we could use to keep track of the arithmetic. But there is absolutely nothing special about this system: any base would give us exactly the same structure to the mathematics For instance numbers will still be prime in any base, they would just happen to have a different label that we give them. All we need from a system of numbers is one that represents as much of the pattern and structure from the inner workings as possible.

Let's look at times tables. In decimal the easiest times tables for people to learn are 1, 2, 5 and 10, which are the factors of the base of the system. In comparison 12 has a lot more factors, namely 1, 2, 3, 4, 6 and 12. In dozenal these much easier than usual. Here are both the 3 and 4 times tables:

Sure you lose 5s being easy, but the number of nice multiplications that you will do in your calculations will be proportionally higher. Similarly when converting fractions into decimals many are quite awkward to do. So ½ and ¼ are fine at 0.5 and 0.25, but even ⅓ has an infinite number of terms in it and that comes up all the time. One way of working these out is to multiply the top and bottom of each fraction by the same thing so that the denominator is a power of 10. E.g. ½=5/10, and then since dividing by 10 is just moving all the digits along past the decimal point to get 0.5.

Let's try the same thing with fractions in dozenals. ½=6/10 by multiplying the top and bottom by 6. Remember that the 10 on the bottom (do) is actually the equivalent of 12. But 6/10=0.6 by just moving the 6 one place past the dozenal point. Similarly, 1/3=4/10=0.4 which is much nicer than 1/3 in decimal. Similarly to the times tables it is all of the fractions with factors of 12 have nice dozenal expansions.

There are other nice traits to this system. Time units have a lot of 12s around with factors of 12 in seconds per minute, minutes per hour, hours per day and months per year. Add in the 360 degrees in a circle and possibly inches in a foot if you insist (although like pence in a shilling this one shouldn't be an issue in the modern world) and you get a lot of inherent units which become easier.

While all serious mathematics wouldn't change at all (because it barely uses any numbers other than 0, 1 or infinity) a switch to dozenal would make mental arithmetic and the sort of maths you learn at primary school much easier. The problem is that there is such a big transition cost of everyone learning maths again that it can't really be done.

There are however some African tribes that use base twelve. Instead of counting using fingers, the cultures that use dozenal tend to count the segments of finger. Look at your open palm and above it you have 4 fingers with 3 segments each. I'll leave you with a neat little dozenal trick. Think about the trick that people do to work out the 9 times table. You hold out all ten fingers and if you wanted to work out 9x3 say, you put down the third finger. To read off the answer you see how many fingers are up before the gap (2) and how many after (7): so we get the correct answer 27.

In dozenal we can do the same with for the E times table. Let's do Ɛx7. Put up your finger segments and mentally cross out the 7th one. In the same way as before our answer is 65 (six do five which is equivalent to 77 in decimal). It turns out we can always do the n-1 times table in base n like this.